Theoretical model of hardness anisotropy in brittle materials

Anisotropy is prominent in the hardness test of single crystals. However, the anisotropic nature is not demonstrated quantitatively in previous hardness model. In this work, it is found that the electron transition energy per unit volume in the glide region and the orientation of glide region play critical roles in determining hardness value and hardness anisotropy for a single crystal material. We express the mathematical definition of hardness anisotropy through simple algebraic relations. The calculated Knoop hardnesses of the single crystals are in good agreement with observations. This theory, extended to polycrystalline materials by including hall-petch effect and quantum size effect, predicts that the polycrystalline diamond with low angle grain boundaries can be harder than single-crystal bulk diamond. Combining first-principles technique and the formula of hardness anisotropy the hardness of monoclinic M-carbon, orthorhombic W-carbon, Z-carbon, and T-carbon are predicted.